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Construction of Symplectic Runge-Kutta Methods for Stochastic Hamiltonian Systems

  • Peng Wang (a1), Jialin Hong (a2) and Dongsheng Xu (a2) (a3)

Abstract

We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems (SHS). Three types of systems, SHS with multiplicative noise, special separable Hamiltonians and multiple additive noise, respectively, are considered in this paper. Stochastic Runge-Kutta (SRK) methods for these systems are investigated, and the corresponding conditions for SRK methods to preserve the symplectic property are given. Based on the weak/strong order and symplectic conditions, some effective schemes are derived. In particular, using the algebraic computation, we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise, and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise, respectively. The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.

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Corresponding author

*Corresponding author. Email addresses: wpemk@163.com; pwang@jlu.edu.cn (P.Wang), hjl@lsec.cc.ac.cn (J. Hong), xuds@lsec.cc.ac.cn (D. Xu)

References

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